Answer :
To write the function for a sinusoid, we need to determine the general form of a sine (or cosine) function, and identify the key parameters such as amplitude, frequency, phase shift, and vertical shift. The general form of a sine function is:
[tex]\[ y = A \sin(B(x - C)) + D \][/tex]
Where:
- [tex]\( A \)[/tex] is the amplitude,
- [tex]\( B \)[/tex] is the frequency,
- [tex]\( C \)[/tex] is the phase shift,
- [tex]\( D \)[/tex] is the vertical shift.
Given:
- A point (0, 1), which means when [tex]\( x = 0 \)[/tex], [tex]\( y = 1 \)[/tex],
- A point [tex]\((\pi, -5)\)[/tex], which means when [tex]\( x = \pi \)[/tex], [tex]\( y = -5 \)[/tex].
Step 1: Determine the vertical shift [tex]\( D \)[/tex]
Since [tex]\( y = 1 \)[/tex] when [tex]\( x = 0 \)[/tex], the vertical shift [tex]\( D \)[/tex] ensures that the average value of the sinusoid is correct.
Similarly, when [tex]\( x = \pi \)[/tex], [tex]\( y = -5 \)[/tex].
First, we calculate the midline, which represents the average value of the maximum and minimum points. We find the midpoint of [tex]\(1\)[/tex] and [tex]\(-5\)[/tex]:
[tex]\[ D = \frac{1 + (-5)}{2} = \frac{-4}{2} = -2 \][/tex]
So, the vertical shift [tex]\( D \)[/tex] is [tex]\(-2\)[/tex].
Step 2: Determine the amplitude [tex]\( A \)[/tex]
The amplitude is the distance from the midline to the maximum or minimum point. Using the points (0, 1) and ([tex]\(\pi\)[/tex], -5):
[tex]\[ A = |1 - (-2)| = |1 + 2| = 3 \][/tex]
Or
[tex]\[ A = |-5 - (-2)| = |-5 + 2| = 3 \][/tex]
Hence, the amplitude [tex]\( A \)[/tex] is 3.
Step 3: Determine the frequency [tex]\( B \)[/tex]
The period of the sinusoid is the distance between two repeating points. A sine wave has a period [tex]\( T \)[/tex]:
Given [tex]\( (0, 1) \)[/tex] and [tex]\( (\pi, -5) \)[/tex], we assume these represent a quarter period from maximum to minimum:
Full period [tex]\( T = 4 \cdot (\text{distance between 0 and } \pi) = 4\pi \)[/tex]
The frequency [tex]\( B \)[/tex] is calculated from the period [tex]\( T \)[/tex]:
[tex]\[ B = \frac{2\pi}{T} = \frac{2\pi}{4\pi} = \frac{1}{2} \][/tex]
So, [tex]\( B \)[/tex] is [tex]\(\frac{1}{2}\)[/tex].
Step 4: Determine the phase shift [tex]\( C \)[/tex]
To determine [tex]\( C \)[/tex], we examine the given points:
At [tex]\( x = 0 \)[/tex], [tex]\( y = 1 \)[/tex]:
[tex]\[ 1 = 3 \sin\left(\frac{1}{2}(0 - C)\right) - 2 \][/tex]
[tex]\[ 3 = 3 \sin\left(-\frac{C}{2}\right) \][/tex]
[tex]\[ \sin\left(-\frac{C}{2}\right) = 1 \][/tex]
Therefore, for [tex]\(\sin(x) = 1\)[/tex]:
[tex]\[ -\frac{C}{2} = \frac{\pi}{2} \][/tex]
[tex]\[ -C = \pi \][/tex]
[tex]\[ C = -\pi \][/tex]
So, the phase shift [tex]\( C \)[/tex] is [tex]\(\pi\)[/tex].
Thus, the equation of the sinusoid is:
[tex]\[ y = 3 \sin\left(\frac{1}{2}(x + \pi)\right) - 2 \][/tex]
This is the function for the given sinusoid.
[tex]\[ y = A \sin(B(x - C)) + D \][/tex]
Where:
- [tex]\( A \)[/tex] is the amplitude,
- [tex]\( B \)[/tex] is the frequency,
- [tex]\( C \)[/tex] is the phase shift,
- [tex]\( D \)[/tex] is the vertical shift.
Given:
- A point (0, 1), which means when [tex]\( x = 0 \)[/tex], [tex]\( y = 1 \)[/tex],
- A point [tex]\((\pi, -5)\)[/tex], which means when [tex]\( x = \pi \)[/tex], [tex]\( y = -5 \)[/tex].
Step 1: Determine the vertical shift [tex]\( D \)[/tex]
Since [tex]\( y = 1 \)[/tex] when [tex]\( x = 0 \)[/tex], the vertical shift [tex]\( D \)[/tex] ensures that the average value of the sinusoid is correct.
Similarly, when [tex]\( x = \pi \)[/tex], [tex]\( y = -5 \)[/tex].
First, we calculate the midline, which represents the average value of the maximum and minimum points. We find the midpoint of [tex]\(1\)[/tex] and [tex]\(-5\)[/tex]:
[tex]\[ D = \frac{1 + (-5)}{2} = \frac{-4}{2} = -2 \][/tex]
So, the vertical shift [tex]\( D \)[/tex] is [tex]\(-2\)[/tex].
Step 2: Determine the amplitude [tex]\( A \)[/tex]
The amplitude is the distance from the midline to the maximum or minimum point. Using the points (0, 1) and ([tex]\(\pi\)[/tex], -5):
[tex]\[ A = |1 - (-2)| = |1 + 2| = 3 \][/tex]
Or
[tex]\[ A = |-5 - (-2)| = |-5 + 2| = 3 \][/tex]
Hence, the amplitude [tex]\( A \)[/tex] is 3.
Step 3: Determine the frequency [tex]\( B \)[/tex]
The period of the sinusoid is the distance between two repeating points. A sine wave has a period [tex]\( T \)[/tex]:
Given [tex]\( (0, 1) \)[/tex] and [tex]\( (\pi, -5) \)[/tex], we assume these represent a quarter period from maximum to minimum:
Full period [tex]\( T = 4 \cdot (\text{distance between 0 and } \pi) = 4\pi \)[/tex]
The frequency [tex]\( B \)[/tex] is calculated from the period [tex]\( T \)[/tex]:
[tex]\[ B = \frac{2\pi}{T} = \frac{2\pi}{4\pi} = \frac{1}{2} \][/tex]
So, [tex]\( B \)[/tex] is [tex]\(\frac{1}{2}\)[/tex].
Step 4: Determine the phase shift [tex]\( C \)[/tex]
To determine [tex]\( C \)[/tex], we examine the given points:
At [tex]\( x = 0 \)[/tex], [tex]\( y = 1 \)[/tex]:
[tex]\[ 1 = 3 \sin\left(\frac{1}{2}(0 - C)\right) - 2 \][/tex]
[tex]\[ 3 = 3 \sin\left(-\frac{C}{2}\right) \][/tex]
[tex]\[ \sin\left(-\frac{C}{2}\right) = 1 \][/tex]
Therefore, for [tex]\(\sin(x) = 1\)[/tex]:
[tex]\[ -\frac{C}{2} = \frac{\pi}{2} \][/tex]
[tex]\[ -C = \pi \][/tex]
[tex]\[ C = -\pi \][/tex]
So, the phase shift [tex]\( C \)[/tex] is [tex]\(\pi\)[/tex].
Thus, the equation of the sinusoid is:
[tex]\[ y = 3 \sin\left(\frac{1}{2}(x + \pi)\right) - 2 \][/tex]
This is the function for the given sinusoid.
